Is it possible to train a neural network with 3 inputs and 12 outputs?
There is nothing stopping you, you can setup Dense Neural Networks to have any size inputs or outputs (simple proof is to imagine a single layer NN with no activation is just a linear transform and given input dim $n$ and output dim $m$, it's just a matrix of $n$ x $m$, trivially this works though with any number of hidden layers)

The better question is should you?. In all honesty, it depends on the data that you have, but, usually, with only 120 examples you'll either overfit completely or do relatively well if the true solution is a very simple function, but, in general, in the common situations where that isn't the case I find myself more likely or not using Bayesian approaches, so I can actually consider confidence (with little data, this is really nice)